Under what conditions is the tensor product of two Noetherian domains also a Noetherian domain? To be more precise about the problem, I am looking at rings of fractions on such tensor products, as a method of generalising the coaction of a Hopf algebra $H$ on an algebra $A$. Thus the coaction on an element of $A$ may lie in a ring of fractions on $A\otimes H$ rather than just $A\otimes H$. Any useful information on this would be gratefully appreciated!